THE COORDINATE PLANE

Axes: 2 perpendicular number lines Origin: where the axes intersect and their zero point x-axis: the horizontal number line y-axis: the vertical number line Quadrants: divide the plane into 4 Coordinate plane: is the plane containing the x- and y-axes x-axis y-axis Quadrant II origin Quadrant III Quadrant IV Quadrant I

x-coordinate: is the first number that corresponds to the numbers on the x-axis y-coordinate: is the second number that corresponds to the numbers on the y-axis Points in the coordinate plane are named by ordered pairs of the form (x,y). The ordered pair for the origin is (0,0)

So how do we find the ordered pair for Z? First follow along the vertical line through Z to get the Z coordinate, on the x-axis. Then go along the horizontal line through Z, to find the y coordinate on the y-axis. The number on the x-axis is 4 and the number on the y-axis is –6. So the ordered pair is (4,-6) Z

A table is often used when working with ordered pairs A table is often used when working with ordered pairs. It is a helpful organizational tool in problem solving. Write ordered pairs for points E,F,G, and H shown at the bottom. Name the quadrant in which each point is located. Use a table to help fine the coordinates of each point. H G F E Point x-coordinate y-coordinate Ordered Pair Quadrant E 4 -2 (4,-2) IV F -5 (0-5) None G -3.5 -4 (-3.5, -4) III H -6 3 (-6,3) II

Plot the following points on a coordinate plane. To graph an ordered pair means to draw a dot at the point on the coordinate plane that corresponds to the ordered pair. This is also called plotting a point. Plot the following points on a coordinate plane. N (-3,-5) K (-4.5,8) L (6,0) x y K N L

Completeness Property for Points in the Plane With the information we have been given we come up with the Completeness Property for Points in the Plane. Exactly one point in the plane is named by a given ordered pair of numbers Exactly one ordered pair of numbers names a given point in the plane. Completeness Property for Points in the Plane

Warm-Up 4 minutes 1) Given that x = 2, find y where y = 5x + 4. 2) Given that x = 1, find y where 2x + 3y = 8.

7-1 Graphing Ordered Pairs Objectives: To plot points using the coordinate system To identify the quadrant associated with a point To identify the coordinates of a point

The Coordinate Plane y x-axis y-axis x origin -12 -10 -8 -6 -4 -2 2 4

Graphing Ordered Pairs ( , ) 4 3 x-coordinate y-coordinate left (negative) right (positive) up (positive) down (negative)

Example 1 Plot the points , , , and . (3,5) (-4,1) (-3,-4) (4,-2) x y -12 -10 -8 -6 -4 -2 2 4 6 8 10 x y (3,5) (-4,1) (4,-2) (-3,-4)

Practice Plot these points on the coordinate plane. 1. (4,6) 2. (6,0) 1. (4,6) 2. (6,0) 3. (-2,5) 4. (-3,-3) 5. (5,-3)

Quadrants y x The point (2,4) is located in the first quadrant. First -12 -10 -8 -6 -4 -2 2 4 6 8 10 x y The point (2,4) is located in the first quadrant. First Quadrant Second Quadrant (2,4) The point (-6,-7) is located in the third quadrant. Third Quadrant Fourth Quadrant (-6,-7)

II I III IV Quadrants y x negative positive positive positive negative ( , ) negative positive ( , ) positive positive x III IV ( , ) negative negative ( , ) positive negative

Practice In which quadrant is each point located? 1. (-3,5) 2. (7,-4) 1. (-3,5) 2. (7,-4) 3. (3,8) 4. (-7,-9) 5. (0,8)

y x Find the coordinates of each point in the graph below. C F D E B A -12 -10 -8 -6 -4 -2 2 4 6 8 10 x y C F D E B A

Drill 1) What quadrant would each point be located in: 2) If you start at the origin and go up 6 and then left 4 units what would be the coordinates of that point?

4.2 Relations Algebra 1

Relation A relation is a set of ordered pairs. Ex: {(2, 3) (3, 6) (-4, 8) (-11, 7)}

Relations Domain: the domain of a set is all the possible x-values. Range: the range of a set is all the possible y-values. Ex: {(2, 4) (3, 6) (8, 1) (-3, -4)} Domain is {2, 3, 8, -3} Range is {4, 6, 1, -4}

Mapping Mapping is when you write the domain in one oval and the range in another oval and then draw arrows from each value in the domain to it’s corresponding value in the range. 4 -1 5 8 6 7 10

Domain: {-3, 0, 2, 3, 4} Range: {-6, -4, 1, 2, 5}

Example What is the domain and range of this graph?

Drill {(3, 4) (-3, 6) (-8, 1) (-5, -4) (2, 4)} { (-5, 4) (3, 6) (4, 7) (-6, 9) (-5, 11)} Find the Domain and Range in each set and state whether or not each one is a function.

Inverse of a Relation To take the inverse of a relation you simply switch the x and y coordinates of each ordered pair. Ex: {(2, 4) (3, 6) (8, 1) (-3, -4)} Inverse is : {(4, 2) (6, 3) (1, 8) (-4, -3)}

Classwork Pages 209 – 210 #’s 18 – 37 #’s 60 – 67

DRILL What is the domain and range for this set? { (2, 3) (3, 8) (-2, 6) (5, 3) (4, 6) ( -2, -5)} 2) Which quadrant is each point in? (-3, 5) b) (-4, -5) c) (5, -2) 3) Solve for y if x = 3 y = 4x – 7

4.3 Equations and Relations October 31st 2008

Using Equations As Relations When you have an equation and a given domain, you simply plug in the x-values to get the corresponding y-values. The y-values you get are your range. Ex: Solve the equation y = 3x + 4 if the domain is {-1, 2, 4, 5} X Y -1 2 4 5

y = -7x + 21 Examples Domain is: { -6, -2, 0, 1, 5, 11} Range is: { } 1 5 11 y = -7x + 21 Domain is: { -6, -2, 0, 1, 5, 11} Range is: { }

Quiz/Class Work Page 214 #’s 6, 7, 10, 14, 15, 20, 21, 32, 33, 52, 53, 54, 55

Warm-Up 5 minutes 1) Determine whether the point (0,3) is a solution to y = 5x + 3. 2) Graph y = -2x + 1

7.3.1 Linear Equations and Their Graphs Objectives: To graph linear equations in two variables

Linear Equations Equations whose graphs are lines are linear equations. Here are some examples: Linear Equations y = 3x + 7 6y = -2 9x – 15y = 7 Nonlinear Equations y = x2 - 4 x2 + y2 = 16 xy = 3

Example 1 Graph the equation 2x + 2y = 6. 2x + 2y = 6 solve for y -2x 3 2 2 1 2 y = 3 - x -2 5 y = 3 - (0) = 3 y = 3 - (1) = 2 y = 3 - (-2) = 5

Example 1 Graph the equation 2x + 2y = 6. x y 3 1 2 -2 5 2x + 2y = 6 8 4 x y 2 3 -8 -6 -4 -2 2 4 6 8 1 2 -2 -2 5 -4 -6 -8

Practice Graph these linear equations using three points. 1) 3y – 12 = 9x 2) 4y + 8 = -16x

Example 2 Graph the equation 3y – 6 = 9x. 3y – 6 = 9x solve for y +6 2 3 3 1 5 y = 3x + 2 -2 -4 y = 3(0) + 2 = 2 y = 3(1) + 2 = 5 y = 3(-2) + 2 = -4

Example 2 Graph the equation 3y – 6 = 9x. x y 2 1 5 -2 -4 3y – 6 = 9x 8 3y – 6 = 9x 6 4 x y 2 2 -8 -6 -4 -2 2 4 6 8 1 5 -2 -2 -4 -4 -6 -8

Practice Graph these linear equations using three points. 1) 6x – 2y = -2 2) -10x – 2y = 8

Homework p.316 #1,3,7,11,15 *Use graph paper for the homework

Warm-Up 6 minutes 1) Graph 4x – 3y = 12 * Get 2 sheets of graph paper and a ruler

7.3.2 Linear Equations and Their Graphs Objectives: To graph linear equations using intercepts

Graphing Using Intercepts The x-intercept of a line is the x-coordinate of the point where the line intercepts the x-axis. The line shown intercepts the x-axis at (2,0). 8 6 4 We say that the x-intercept is 2. 2 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8

Graphing Using Intercepts The y-intercept of a line is the y-coordinate of the point where the line intercepts the y-axis. The line shown intercepts the y-axis at (0,-6). 8 6 We say that the y-intercept is -6. 4 2 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8

Example 1 Graph 4x – 3y = 12 using intercepts. *To find the y-intercept, let x = 0. 4x – 3y = 12 x y 4(0) – 3y = 12 -4 0 – 3y = 12 -3y = 12 -3 -3 y = -4

Example 1 Graph 4x – 3y = 12 using intercepts. *To find the x-intercept, let y = 0. 4x – 3y = 12 x y 4x – 3(0) = 12 -4 4x - 0 = 12 3 4x = 12 4 4 x = 3

Example 1 Graph 4x – 3y = 12 using intercepts. x y -4 3 8 6 4 2 -8 -6 -4 -8 -6 -4 -2 2 4 6 8 -2 3 -4 -6 -8

Example 2 Graph 2x + 5y = 10 using intercepts. *To find the y-intercept, let x = 0. 2x + 5y = 10 x y 2(0) + 5y = 10 2 0 + 5y = 10 5y = 10 5 5 y = 2

Example 2 Graph 2x + 5y = 10 using intercepts. *To find the x-intercept, let y = 0. 2x + 5y = 10 x y 2x + 5(0) = 10 2 2x + 0 = 10 5 2x = 10 2 2 x = 5

Example 2 Graph 2x + 5y = 10 using intercepts. x y 2 5 8 6 4 2 -8 -6 2 -8 -6 -4 -2 2 4 6 8 -2 5 -4 -6 -8

Practice Graph using intercepts. 1) 5x + 7y = 35 2) 8x + 2y = 24

Homework p.316 #17,20,21,23,26,27 *Use graph paper for the homework

4 minutes Warm-Up Determine the coordinates of each point in the graph below. -12 -10 -8 -6 -4 -2 2 4 6 8 10 x y A B C D

7-2 Graphing Equations Objectives: To determine whether an ordered pair is a solution of an equation To graph equations in two variables

Solutions of Equations How many solutions does the equation 4x + 6 = 14 have? What are they? 4x = 8 4 4 x = 2 An equation such as y = 3x + 7 has many solutions, which we write as ordered pairs of numbers. (x,y)

Example 1 Determine whether is a solution of y = 3x -2. ( , ) 2 4 ( , ) 2 4 y = 3x - 2 (4) = 3 (2) - 2 4 = 6 - 2 4 = 4 (2,4) is a solution of y = 3x -2

Practice 1) Determine whether (2,3) is a solution of y = 2x + 3.

Example 2 Find three solutions of y = 2x + 11. x y y = 2x + 11 11 y = 2(-1) + 11 y = 13 y = 11 y = 9 1 13 -1 9 *Our three solutions are (0,11), (1,13), and (-1,9).

Example 3 Graph the equation 6x + 2y = 4. 6x + 2y = 4 solve for y -6x 2 2 2 1 -1 y = 2 – 3x -2 8 y = 2 - 3(0) = 2 y = 2 - 3(1) = -1 y = 2 - 3(-2) = 8

Example 3 Graph the equation 6x + 2y = 4. x y 2 1 -1 -2 8 6x + 2y = 4 2 -8 -6 -4 -2 2 4 6 8 1 -1 -2 -2 8 -4 -6 -8

Practice Graph each equation. 1) y – x = 3 2) y = x

Homework p.311 #3,5,7,11,23,27

Warm-Up 10 minutes Graph these equations: -x + 2y = 4 2x + 3y = 8

7.3.3 Linear Equations and Their Graphs Objectives: To graph linear equations that graph as horizontal and vertical lines

Graphing Horizontal and Vertical Lines The standard form of a linear equation in two variables is Ax + By = C, where A,B, and C are constants and A and B are not both 0. 3x + 4y = 12 6x + 7y = 23

Example 1 Graph y = -2. write the equation in standard form Ax + By = C (0)x + (1)y = -2 -8 -6 -4 -2 2 4 6 8 for any value of x y = -2

Example 2 Graph x = 7. write the equation in standard form Ax + By = C (1)x + (0)y = 7 -8 -6 -4 -2 2 4 6 8 x = 7 for any value of y

Practice Graph these equations. 1) x = 5 2) y = -4 3) x = 0

Homework p.316 #11,13,19,25,27,35,37,41 *Use graph paper for the homework

DRILL Find the next three terms in each pattern below. 1. 3, 6, 9, 12, … 2. 45, 39, 33, 27, … 3. 4, 5, 8, 13, 20, … 4. –12, –7, –2, 3, … 5. 1, 2, 4, 7, … 6. –2, –1.75, –1.5, –1.25, …

4.7 Arithmetic Sequences Objective:

Examples